Lect25_OrbitalAngMom - Lecture 25: Orbital Angular Momentum...

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Unformatted text preview: Lecture 25: Orbital Angular Momentum PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore The General Theory of Angular Momentum • Starting point: – Assume you have three operators that satisfy the commutation relations: – Let: • Conclusions: – Simultaneous eigenstates of J 2 and J z exist – They must satisfy: – Where the quantum numbers take on the values: y x z y x iJ J J J J J J ± = + + = ± 2 2 2 2 y x z x z y z y x J i J J J i J J J i J J h h h = = = ] , [ ] , [ ] , [ m j m m j J m j j j m j J z , , , ) 1 ( , 2 2 h h = + = 1 , ) 1 ( ) 1 ( , ± ± ! + = ± m j m m j j m j J h j j j j j m j , 1 , , 2 , 1 , , 3 , 2 5 , 2 , 2 3 , 1 , 2 1 , ! + ! + ! ! = = K K Orbital Angular Momentum • For orbital angular momentum we have: • So that: • In coordinate representation we have: P R L r r v ! = x y z YP XP L ! = v L "# i h r r $ r % ( ) ! " " ! " # # # # # # = $ % sin 1 1 det r r r r e e e r r r r r r r ! " ! " ! # # + # # $ = e e r r sin 1 Continued • Note that it doesn’t depend on r or ! / ! r ....
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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Lect25_OrbitalAngMom - Lecture 25: Orbital Angular Momentum...

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